{
  "abstract": "The lampshuffler group of $\\mathbb{Z}$ is the semidirect product $\\operatorname{FSym}(\\mathbb{Z}) \\rtimes \\mathbb{Z}$, which consists of all permutations of $\\mathbb{Z}$ that act as a translation outside a finite set. This infinite permutation group naturally contains as subgroups the wreath products $H \\wr \\mathbb{Z}$ for every finite group $H$. We prove that the Subgroup Membership Problem, and more generally, the Submonoid Membership Problem, are decidable in $\\operatorname{FSym}(\\mathbb{Z}) \\rtimes \\mathbb{Z}$. Our proof reduces Subgroup and Submonoid Membership in $\\operatorname{FSym}(\\mathbb{Z}) \\rtimes \\mathbb{Z}$ to Subgroup Membership in the wreath products $H \\wr \\mathbb{Z}$, which was shown to be decidable by Lohrey, Steinberg and Zetzsche (2015).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06108",
  "author_total": 2,
  "authors": [
    {
      "name": "Corentin Bodart"
    },
    {
      "name": "Ruiwen Dong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a919xv1cp9fq4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06108",
  "fulltext_url": null,
  "id": "a919xv1cp9fq4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06108",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7220700721",
  "pdf_url": "https://arxiv.org/pdf/2610.06108",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a919xv1cp9fq4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06108",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06108"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Subgroup and Submonoid Membership in the lampshuffler of $\\mathbb{Z}$",
  "title_text": "Subgroup and Submonoid Membership in the lampshuffler of ℤ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/a919xv1cp9fq4",
  "venue": "arXiv",
  "versions": []
}